jmc

The units digit of a power of an integer is determined by the units digit of the integer; that is, the tens digit, hundreds digit, etc... of the integer have no effect on the units digit of the result. In this problem, the units digit of $19^{19}$ is the units digit of $9^{19}$. Note that $9^1=9$ ends in 9, $9^2=81$ ends in 1, $9^3=729$ ends in 9, and, in general, the units digit of odd powers of 9 is 9, whereas the units digit of even powers of 9 is 1. Since both exponents are odd, the sum of their units digits is $9+9=18$, the units digit of which is $\boxed{8}.$